Embodied cognition,embodied regulation,and the Data Rate
Theorem
Rodrick Wallace
Division of Epidemiology
The New York State Psychiatric Institute
January 23,2014
Abstract
The Data Rate Theorem that establishes a formal linkage
between linear control theory and information theory carries
deep implications for the design of biologically inspired cogni
tive architectures (BICA),and for the more general study of
embodied cognition.For example,modest extensions of the
theorem provide a spectrum of necessary conditions dynamic
statistical models that will be useful in empirical studies.A
large deviations argument,however,suggests that the stabi
lization of such systems is itself an interpenetrating dynamic
process necessarily convoluted with embodied cognition.As
our experience with mental disorders and chronic disease im
plies,evolutionary process has had only modest success in
the regulation and control of cognitive biological phenomena.
For humans,the central role of culture has long been known.
Although a groundstate collapse analogous to generalized
anxiety appears ubiquitous to such systems,lack of cultural
modulation for realtime automatons or distributed cognition
manmachine`cockpits'makes them particularly subject to a
canonical pathology under which`all possible targets are ene
mies'.More general dysfunctions of largescale topology and
connectivity analogous to autism spectrum and schizopheno
form disorders also appear likely.A kind of machine psy
chiatry may become a central engineering discipline as the
number of computation cores in realtime critical systems in
creases exponentially over the next few decades.
Key words:articial intelligence;control;information
theory,language;machine psychiatry;regulation
...[O]ur rst move is simply to treat perception
action problems and language problems as the same
kind of thing...Linguistic information is a task re
source in exactly the same way as perceptual infor
mation...Our behavior emerges from a pool of po
tential task resources that include the body,the en
vironment and...the brain.
{ A.D.Wilson and S.Golonka,2013
Box 47,NYSPI,1051 Riverside Dr.,NY,NY,10032,USA.Wal
lace@nyspi.columbia.edu,rodrick.wallace@gmail.com
1 Introduction
According to Samsonovitch (2012),
The BICA Challenge is the challenge to create a
generalpurpose,reallife computational equivalent
of the human mind using an approach based on bio
logically inspired cognitive architectures...To solve
it,we need to understand at a computational level
how natural intelligent systems develop their cog
nitive,metacognitive and learning functions.The
solution is expected to lead us to a breakthrough to
intelligent agents integrated into the human society
as its members.
Natural cognitive systems operate at all scales and levels of
organization of biological process (e.g.,Wallace,2012,2014).
The failure of low level biological cognition in humans is often
expressed through early onset of the intractable chronic dis
eases of senescence (e.g.,Wallace and Wallace,2010,2013).
Failure of high order cognition in humans has been the sub
ject of intensive scientic study for over two hundred years,
with little if any consensus.As JohnsonLaird et al.(2006)
put it,
Current knowledge about psychological illnesses
is comparable to the medical understanding of epi
demics in the early 19th Century.Physicians real
ized that cholera,for example,was a specic disease,
which killed about a third of the people whom it in
fected.What they disagreed about was the cause,
the pathology,and the communication of the disease.
Similarly,most medical professionals these days re
alize that psychological illnesses occur...but they
disagree about their cause and pathology.
And as the press chatter surrounding the release of the lat
est ocial US nosology of mental disorders { the socalled
`DSMV'{ indicates,this may be something an understate
ment.Indeed,the entire enterprise of the Diagnostic and Sta
tistical Manual of Mental Disorders has been characterized as
`prescientic'(e.g.,Gilbert,2001).Atmanspacher (2006),for
example,argues that formal theory of highlevel cognition is
itself at a point like that of physics 400 years ago,with the
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basic entities and the relations between them yet to be deter
mined.Further complications arise via the overwhelming in
uence of culture on both mental process and its dysfunction
(e.g.,Heine,2001;Kleinman and Cohen 1997),something to
which we will eventually return.See Chapter 5 of Wallace and
Wallace (2013) for a more detailed summary of these matters.
The overall inference is that stabilization and regulation of
high order cognition may be as dicult as the BICAChallenge
itself.
Varela,Thompson and Rosch (1991),in their study The
Embodied Mind:Cognitive Science and Human Experience,
asserted that the world is portrayed and determined by mu
tual interaction between the physiology of an organism,its
sensimotor circuitry,and the environment.The essential
point,in their view,being the inherent structural coupling
of brainbodyworld.Lively debate has followed and con
tinues (e.g.,Clark,1998;M.Wilson,2002;A.Wilson and
S.Golonka,2013).Brooks (1986),and many others,have
explored and extended analogous ideas,particularly focus
ing on robotics.It is possible to make a basic approach to
these problems via the Data Rate Theorem,and to include
as well regulation and stabilization mechanisms in a unitary
construct that must interpenetrate in a similar manner.
Cognition can be described in terms of a sophisticated real
time feedback between interior and exterior,necessarily con
strained,as Dretske (1994) has noted,by certain asymptotic
limit theorems of probability:
Communication theory can be interpreted as
telling one something important about the condi
tions that are needed for the transmission of infor
mation as ordinarily understood,about what it takes
for the transmission of semantic information.This
has tempted people...to exploit [information theory]
in semantic and cognitive studies...
...Unless there is a statistically reliable channel of
communication between [a source and a receiver]...
no signal can carry semantic information...[thus]
the channel over which the [semantic] signal arrives
[must satisfy] the appropriate statistical constraints
of information theory.
Recent intersection of that theory with the formalisms of
realtime feedback systems { control theory { may provide
insight into matters of embodied cognition and the parallel
synergistic problemof embodied regulation and control.Here,
we extend that work and apply the resulting conceptual model
toward formally characterizing the unitary structural coupling
of brainbodyworld.In the process,we will explore dynamic
statistical models that can be tted to data.
2 The DataRate Theorem
The recentlyformalized datarate theorem,a generalization
of the classic Bode integral theorem for linear control systems
(e.g.,Yu and Mehta,2010;Kitano,2007;Csete and Doyle,
2002),describes the stability of linear feedback control under
data rate constraints (e.g.,Mitter,2001;Tatikonda and Mit
ter,2004;Sahai,2004;Sahai and Mitter,2006;Minero et al.,
2009;Nair et al.,2007;You and Xie,2013).Given a noisefree
data link between a discrete linear plant and its controller,
unstable modes can be stabilized only if the feedback data
rate H is greater than the rate of`topological information'
generated by the unstable system.For the simplest incarna
tion,if the linear matrix equation of the plant is of the form
x
t+1
= Ax
t
+:::,where x
t
is the ndimensional state vector
at time t,then the necessary condition for stabilizability is
H > log[jdetA
u
j] (1)
where det is the determinant and A
u
is the decoupled unstable
component of A,i.e.,the part having eigenvalues 1.
The essential matter is that there is a critical positive data
rate below which there does not exist any quantization and
control scheme able to stabilize an unstable (linear) feedback
system.
This result,and its variations,are as fundamental as the
Shannon Coding and Source Coding Theorems,and the Rate
Distortion Theorem (Cover and Thomas,2006;Ash,1990;
Khinchin,1957).
We will entertain and extend these considerations,using
methods from cognitive theory to explore brainbodyworld
dynamics that inherently take place under datarate con
straints.
The essential analytic tool will be something much like Pet
tini's (2007)`topological hypothesis'{ a version of Landau's
spontaneous symmetry breaking insight for physical systems
(Landau and Lifshitz,2007) { which infers that punctuated
events often involve a change in the topology of an underly
ing conguration space,and the observed singularities in the
measures of interest can be interpreted as a`shadow'of major
topological change happening at a more basic level.
The preferred tool for the study of such topological changes
is Morse Theory (Pettini,2007;Matsumoto,2002),summa
rized in the Mathematical Appendix,and we shall construct
a relevant Morse Function as a`representation'of the under
lying theory.
We begin with recapitulation of an approach to cognition
using the asymptotic limit theorems of information theory
(Wallace 2000,2005a,b,2007,2012,2014).
3 Cognition as an information
source
Atlan and Cohen (1998) argue that the essence of cognition
involves comparison of a perceived signal with an internal,
learned or inherited picture of the world,and then choice of
one response from a much larger repertoire of possible re
sponses.That is,cognitive pattern recognitionandresponse
proceeds by an algorithmic combination of an incoming exter
nal sensory signal with an internal ongoing activity { incor
porating the internalized picture of the world { and triggering
an appropriate action based on a decision that the pattern of
sensory activity requires a response.
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Incoming sensory input is thus mixed in an unspecied but
systematic manner with internal ongoing activity to create a
path of combined signals x = (a
0
;a
1
;:::;a
n
;:::).Each a
k
thus
represents some functional composition of the internal and
the external.An application of this perspective to a standard
neural network is given in Wallace (2005a,p.34).
This path is fed into a highly nonlinear,but otherwise sim
ilarly unspecied,decision function,h,generating an output
h(x) that is an element of one of two disjoint sets B
0
and B
1
of possible system responses.Let
B
0
fb
0
;:::;b
k
g;
B
1
fb
k+1
;:::;b
m
g:
Assume a graded response,supposing that if
h(x) 2 B
0
;
the pattern is not recognized,and if
h(x) 2 B
1
;
the pattern is recognized,and some action b
j
;k +1 j m
takes place.
Interest focuses on paths x triggering pattern recognition
andresponse:given a xed initial state a
0
,examine all possi
ble subsequent paths x beginning with a
0
and leading to the
event h(x) 2 B
1
.Thus h(a
0
;:::;a
j
) 2 B
0
for all 0 j < m,
but h(a
0
;:::;a
m
) 2 B
1
.
For each positive integer n,take N(n) as the number of high
probability paths of length n that begin with some particular
a
0
and lead to the condition h(x) 2 B
1
.Call such paths
`meaningful',assuming that N(n) will be considerably less
than the number of all possible paths of length n leading from
a
0
to the condition h(x) 2 B
1
.
Identication of the`alphabet'of the states a
j
;B
k
may de
pend on the proper system coarse graining in the sense of
symbolic dynamics (e.g.,Beck and Schlogl,1993).
Combining algorithm,the form of the function h,and the
details of grammar and syntax,are all unspecied in this
model.The assumption permitting inference on necessary
conditions constrained by the asymptotic limit theorems of
information theory is that the nite limit
H lim
n!1
log[N(n)]
n
both exists and is independent of the path x.Again,N(n) is
the number of high probability paths of length n.
Call such a pattern recognitionandresponse cognitive pro
cess ergodic.Not all cognitive processes are likely to be er
godic,implying that H,if it indeed exists at all,is path de
pendent,although extension to nearly ergodic processes,in a
certain sense,seems possible (e.g.,Wallace,2005a,pp.3132).
Invoking the ShannonMcMillan Theorem (Cover and
Thomas,2006;Khinchin,1957),we take it possible to de
ne an adiabatically,piecewise stationary,ergodic infor
mation source X associated with stochastic variates X
j
having joint and conditional probabilities P(a
0
;:::;a
n
) and
P(a
n
ja
0
;:::;a
n1
) such that appropriate joint and conditional
Shannon uncertainties satisfy the classic relations
H[X] = lim
n!1
log[N(n)]
n
=
lim
n!1
H(X
n
jX
0
;:::;X
n1
) =
lim
n!1
H(X
0
;:::;X
n
)
n
(2)
This information source is dened as dual to the underly
ing ergodic cognitive process,in the sense of Wallace (2005a,
2007).
`Adiabatic'means that,when the information source is
properly parameterized,within continuous`pieces',changes
in parameter values take place slowly enough so that the in
formation source remains as close to stationary and ergodic as
needed to make the fundamental limit theorems work.`Sta
tionary'means that probabilities do not change in time,and
`ergodic'that crosssectional means converge to longtime av
erages.Between pieces it is necessary to invoke phase change
formalism,a`biological'renormalization that generalizes Wil
son's (1971) approach to physical phase transition (Wallace,
2005a).
Shannon uncertainties H(:::) are crosssectional lawof
largenumbers sums of the form
P
k
P
k
log[P
k
],where the
P
k
constitute a probability distribution.See Cover and
Thomas (2006),Ash (1990),or Khinchin (1957) for the stan
dard details.
For cognitive systems,an equivalence class algebra can be
constructed by choosing dierent origin points a
0
,and den
ing the equivalence of two states a
m
;a
n
by the existence of
high probability meaningful paths connecting them to the
same origin point.Disjoint partition by equivalence class,
analogous to orbit equivalence classes for a dynamical sys
tem,denes the vertices of a network of cognitive dual lan
guages that interact to actually constitute the system of in
terest.Each vertex then represents a dierent information
source dual to a cognitive process.This is not a representa
tion of a network of interacting physical systems as such,in
the sense of network systems biology (e.g.,Arrell and Terzic,
2010).It is an abstract set of languages dual to the set of
cognitive processes of interest,that may become linked into
higher order structures.
Topology,in the 20th century,became an object of alge
braic study,socalled algebraic topology,via the fundamental
underlying symmetries of geometric spaces.Rotations,mir
ror transformations,simple (`ane') displacements,and the
like,uniquely characterize topological spaces,and the net
works inherent to cognitive phenomena having dual informa
tion sources also have complex underlying symmetries:char
acterization via equivalence classes denes a groupoid,an ex
tension of the idea of a symmetry group,as summarized by
Brown (1987) and Weinstein (1996).Linkages across this set
of languages occur via the groupoid generalization of Landau's
spontaneous symmetry breaking arguments that will be used
below (Landau and Lifshitz,2007;Pettini,2007).See the
Mathematical Appendix for a brief summary of basic mate
rial on groupoids.
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It is important to recognize that we are not constrained to
the AtlanCohen model of cognition.The essential inference
is that a broad class of cognitive phenomena can be associated
with a dual information source.That is,cognition inevitably
involves choice,choice reduces uncertainty,and this implies
the existence of an information source.
Extension to nonergodic information sources can be done
using the methods of Wallace (2005a,Sec.3.1).
4 Environment as an information
source
Multifactorial cognitive and behavioral systems interact with,
aect,and are aected by,embedding environments that`re
member'interaction by various mechanisms.It is possible to
reexpress environmental dynamics in terms of a grammar and
syntax that represent the output of an information source {
another generalized language.
Some examples:
1.The turnofthe seasons in a temperate climate,for many
ecosystems,looks remarkably the same year after year:the
ice melts,the migrating birds return,the trees bud,the grass
grows,plants and animals reproduce,high summer arrives,
the foliage turns,the birds leave,frost,snow,the rivers freeze,
and so on.
2.Human interactions take place within fairly well dened
social,cultural,and historical constraints,depending on con
text:birthday party behaviors are not the same as cocktail
party behaviors in a particular social set,but both will be
characteristic.
3.Gene expression during development is highly patterned
by embedding environmental context via`norms of reaction'
(e.g.,Wallace and Wallace,2010).
Suppose it possible to coarsegrain the generalized`ecosys
tem'at time t,in the sense of symbolic dynamics (e.g.,Beck
and Schlogl,1993) according to some appropriate partition of
the phase space in which each division A
j
represent a partic
ular range of numbers of each possible fundamental actor in
the generalized ecosystem,along with associated larger sys
tem parameters.What is of particular interest is the set of
longitudinal paths,system statements,in a sense,of the form
x(n) = A
0
;A
1
;:::;A
n
dened in terms of some natural time
unit of the system.Thus n corresponds to an again appropri
ate characteristic time unit T,so that t = T;2T;:::;nT.
Again,the central interest is in serial correlations along
paths.
Let N(n) be the number of possible paths of length n that
are consistent with the underlying grammar and syntax of the
appropriately coarsegrained embedding ecosystem,in a large
sense.As above,the fundamental assumptions are that { for
this chosen coarsegraining { N(n),the number of possible
grammatical paths,is much smaller than the total number of
paths possible,and that,in the limit of (relatively) large n,
H = lim
n!1
log[N(n)]=n both exists and is independent of
path.
These conditions represent a parallel with parametric
statistics systems for which the assumptions are not true will
require specialized approaches.
Nonetheless,not all possible ecosystemcoarsegrainings are
likely to work,and dierent such divisions,even when appro
priate,might well lead to dierent descriptive quasilanguages
for the ecosystemof interest.Thus,empirical identication of
relevant coarsegrainings for which this theory will work may
represent a dicult scientic problem.
Given an appropriately chosen coarsegraining,dene joint
and conditional probabilities for dierent ecosystem paths,
having the form P(A
0
;A
1
;:::;A
n
),P(A
n
jA
0
;:::;A
n1
),such
that appropriate joint and conditional Shannon uncertainties
can be dened on them that satisfy equation (2).
Taking the denitions of Shannon uncertainties as above,
and arguing backwards from the latter two parts of equation
(2),it is indeed possible to recover the rst,and divide the set
of all possible ecosystemtemporal paths into two subsets,one
very small,containing the grammatically correct,and hence
highly probable paths,that we will call`meaningful',and a
much larger set of vanishingly low probability.
5 Body dynamics as an information
source
Body movement is inherently constrained by evolutionary
Bauplan:snakes do not brachiate,humans cannot (easily)
scratch their ears with their hind legs,sh do not breathe air,
nor mammals water.This is so evident that one simply does
not think about it.Nonetheless,teaching a human to walk
and talk,a bird to y,or a lion to hunt,in spite of evolu
tion,are arduous enterprises that take considerable attention
from parents or even larger social groupings.Given the basic
bodyplan of head and four limbs,or two feet and wings,or of
a limbless spine,the essential point is that not all motions are
possible.Bauplan imposes limits on dynamics.That is,if we
coarsegrain motions,perhaps using some formof the standard
methods for choreography transcription appropriate to the or
ganism (or mechanism) under study,we see immediately that
not all`statements'possible using the dance symbols have the
same probability.That is,there will inevitably be a grammar
and syntax to observed bodybased behaviors imposed by evo
lutionary or explicit design bauplan.Sequences of symbols,
say of length n,representing observed motions can be segre
gated into two sets,the rst,and vastly larger,consisting of
meaningless sequences (like humans scratching their ears with
their feet) that have vanishingly small probability as n!1.
The other set,consistent with underlying bauplan grammar
and syntax,can be viewed as the output of an information
source,in precisely the manner of the previous two sections,
in rst approximation following the relations of equation (2).
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6 Interacting information sources
Given a set of information sources that are linked to solve a
problem,in the sense of Wilson and Golonka (2013),the`no
free lunch'theorem (English,1996;Wolpert and Macready,
1995,1997) extends a network theorybased theory (e.g.,Ar
rell and Terzic,2010).Wolpert and Macready show there ex
ists no generally superior computational function optimizer.
That is,there is no`free lunch'in the sense that an optimizer
pays for superior performance on some functions with inferior
performance on others gains and losses balance precisely,and
all optimizers have identical average performance.In sum,
an optimizer has to pay for its superiority on one subset of
functions with inferiority on the complementary subset.
This result is wellknown using another description.Shan
non (1959) recognized a powerful duality between the prop
erties of an information source with a distortion measure and
those of a channel.This duality is enhanced if we consider
channels in which there is a cost associated with the dierent
letters.Solving this problem corresponds to nding a source
that is right for the channel and the desired cost.Evaluat
ing the rate distortion function for a source corresponds to
nding a channel that is just right for the source and allowed
distortion level.
Yet another approach to the same result is the through the
`tuning theorem'(Wallace,2005a,Sec.2.2),which inverts the
Shannon Coding Theorem by noting that,formally,one can
view the channel as`transmitted'by the signal.Then a dual
channel capacity can be dened in terms of the channel prob
ability distribution that maximizes information transmission
assuming a xed message probability distribution.
From the no free lunch argument,Shannon's insight,or the
`tuning theorem',it becomes clear that dierent challenges
facing any cognitive system,distributed collection of them,
or interacting set of other information sources,that consti
tute an organism or automaton,must be met by dierent
arrangements of cooperating modules represented as infor
mation sources.
It is possible to make a very abstract picture of this phe
nomenon based on the network of linkages between the in
formation sources dual to the individual`unconscious'cogni
tive modules (UCM),and those of related information sources
with which they interact.That is,a shifting,taskmapped,
network of information sources is continually reexpressed:
given two distinct problems classes confronting the organism
or automaton,there must be two dierent wirings of the infor
mation sources,including those dual to the available UCM,
with the network graph edges measured by the amount of
information crosstalk between sets of nodes representing the
dierent sources.
Thus fully embodied systems,in the sense of Wilson and
Golonka (2013),involve interaction between very general sets
of information sources assembled into a`taskspecic device'
in the sense of Bingham (1988) that is necessarily highly tun
able.This mechanism represents a broad evolutionary gen
eralization of the`shifting spotlight'characterizing the global
neuronal workspace model of consciousness (Wallace,2005a).
We will return to this point in more detail below.
The mutual information measure of crosstalk is not inher
ently xed,but can continuously vary in magnitude.This
suggests a parameterized renormalization:the modular net
work structure linked by crosstalk has a topology depending
on the degree of interaction of interest.
Dene an interaction parameter!,a real positive number,
and look at geometric structures dened in terms of linkages
set to zero if mutual information is less than,and`renormal
ized'to unity if greater than,!.Any given!will dene
a regime of giant components of network elements linked by
mutual information greater than or equal to it.
Now invert the argument:a given topology for the giant
component will,in turn,dene some critical value,!
C
,so that
network elements interacting by mutual information less than
that value will be unable to participate,i.e.,will be locked out
and not be consciously or otherwise perceived.See Wallace
(2005a,2012) for details.Thus!is a tunable,syntactically
dependent,detection limit that depends critically on the in
stantaneous topology of the giant component of linked infor
mation sources dening the analog to a global broadcast of
consciousness.That topology is the basic tunable syntactic
lter across the underlying modular structure,and variation
in!is only one aspect of more general topological properties
that can be described in terms of index theorems,where far
more general analytic constraints can become closely linked
to the topological structure and dynamics of underlying net
works,and,in fact,can stand in place of them (Atyah and
Singer,1963;Hazewinkel,2002).
7 Simple regulation
Continuing the formal theory,information sources are often
not independent,but are correlated,so that a joint infor
mation source { representing,for example,the interaction
between brain,body,and the environment { can be dened
having the properties
H(X
1
;:::;X
n
)
n
X
j=1
H(X
j
) (3)
with equality only for isolated,independent information
streams.
This is the information chain rule (Cover and Thomas,
2006),and has implications for free energy consumption in
regulation and control of embodied cognitive processes.Feyn
man (2000) describes how information and free energy have
an inherent duality,dening information precisely as the free
energy needed to erase a message.The argument is quite di
rect,and it is easy to design an idealized machine that turns
the information within a message directly into usable work {
free energy.Information is a form of free energy and the con
struction and transmission of information within living things
{ the physical instantiation of information { consumes con
siderable free energy,with inevitable { and massive { losses
via the second law of thermodynamics.
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Suppose an intensity of available free energy is associated
with each dened joint and individual information source
H(X;Y );H(X);H(Y ),e.g.,rates M
X;Y
,M
X
,M
Y
.
Although information is a form of free energy,there is nec
essarily a massive entropic loss in its actual expression,so
that the probability distribution of a source uncertainty H
might be written in Gibbs form as
P[H]
exp[H=M]
R
exp[H=M]dH
(4)
assuming is very small.
To rst order,then,
^
H
Z
HP[H]dH M (5)
and,using equation (3),
^
H(X;Y )
^
H(X) +
^
H(Y )
M
X;Y
M
X
+M
Y
(6)
Thus,as a consequence of the information chain rule,al
lowing crosstalk consumes a lower rate of free energy than
isolating information sources.That is,in general,it takes
more free energy { higher total cost { to isolate a set of cogni
tive phenomena and an embedding environment than it does
to allow them to engage in crosstalk (Wallace,2012).
Hence,at the free energy expense of supporting two infor
mation sources,{ X and Y together { it is possible to catalyze
a set of joint paths dened by their joint information source.
In consequence,given a cognitive module (or set of them)
having an associated information source H(:::),an external
information source Y { the embedding environment { can
catalyze the joint paths associated with the joint information
source H(:::;Y ) so that a particular chosen developmental or
behavioral pathway { in a large sense { has the lowest relative
free energy.
At the expense of larger global free information expenditure
{ maintaining two (or more) information sources with their
often considerable entropic losses instead of one { the system
can feed,in a sense,the generalized physiology of a Maxwell's
Demon,doing work so that environmental signals can direct
system cognitive response,thus locally reducing uncertainty
at the expense of larger global entropy production.
Given a cognitive biological system characterized by an in
formation source X,in the context of { for humans { an ex
plicitly,slowlychanging,cultural`environmental'information
source Y,we will be particularly interested in the joint source
uncertainty dened as H(X;Y ),and next examine some de
tails of how such a mutually embedded system might operate
in real time,focusing on the role of rapidlychanging feedback
information,via the Data Rate Theorem.
8 Phase transition
A fundamental homology between the information source un
certainty dual to a cognitive process and the free energy den
sity of a physical system arises,in part,from the formal simi
larity between their denitions in the asymptotic limit.Infor
mation source uncertainty can be dened as in the rst part
of equation (2).This is quite analogous to the free energy
density of a physical system in terms of the thermodynamic
limit of innite volume (e.g.,Wilson,1971;Wallace,2005a).
Feynman (2000) provides a series of physical examples,based
on Bennett's (1988) work,where this homology is an identity,
at least for very simple systems.Bennett argues,in terms
of idealized irreducibly elementary computing machines,that
the information contained in a message can be viewed as the
work saved by not needing to recompute what has been trans
mitted.
It is possible to model a cognitive system interacting with
an embedding environment using a simple extension of the
languageofcognition approach above.Recall that cognitive
processes can be formally associated with information sources,
and how a formal equivalence class algebra can be constructed
for a complicated cognitive system by choosing dierent ori
gin points in a particular abstract`space'and dening the
equivalence of two states by the existence of a high probabil
ity meaningful path connecting each of them to some dened
origin point within that space.
Recall that disjoint partition by equivalence class is analo
gous to orbit equivalence relations for dynamical systems,and
denes the vertices of a network of cognitive dual languages
available to the system:each vertex represents a dierent in
formation source dual to a cognitive process.The structure
creates a large groupoid,with each orbit corresponding to a
transitive groupoid whose disjoint union is the full groupoid,
and each subgroupoid associated with its own dual informa
tion source.Larger groupoids will,in general,have`richer'
dual information sources than smaller.
We can now begin to examine the relation between system
cognition and the feedback of information from the rapidly
changing realtime (as opposed to a slowtime cultural or
other) environment,H,in the sense of equation (1).
With each subgroupoid G
i
of the (large) cognitive groupoid
we can associate a joint information source uncertainty
H(X
G
i
;Y ) H
G
i
,where X is the dual information source of
the cognitive phenomenon of interest,and Y that of the em
bedding environmental context { largely dened,for humans,
in terms of culture and pathdependent historical trajectory.
Real time dynamic responses of a cognitive systemcan now
be represented by high probability paths connecting`initial'
multivariate states to`nal'congurations,across a great va
riety of beginning and end points.This creates a similar va
riety of groupoid classications and associated dual cognitive
processes in which the equivalence of two states is dened by
linkages to the same beginning and end states.Thus,we will
show,it becomes possible to construct a`groupoid free energy'
driven by the quality of rapidlychanging,realtime informa
tion coming from the embedding ecosystem,represented by
the information rate H,taken as a temperature analog.
For humans in particular,H is an embedding context for
the underlying cognitive processes of interest,here the tun
able,shifting,global broadcasts of consciousness as embedded
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in,and regulated by,culture.The argumentbyabduction
from physical theory is,then,that H constitutes a kind of
thermal bath for the processes of culturallychanneled cogni
tion.Thus we can,in analogy with the standard approach
from physics (Pettini,2007;Landau and Lifshitz,2007) con
struct a Morse Function by writing a pseudoprobability for
the jointlydened information sources X
G
i
;Y having source
uncertainty H
G
i
as
P[H
G
i
] =
exp[H
G
i
=H)]
P
j
exp[H
G
j
=H]
(7)
where is an appropriate dimensionless constant characteris
tic of the particular system.The sum is over all possible sub
groupiods of the largest available symmetry groupoid.Again,
compound sources,formed by the (tunable,shifting) union
of underlying transitive groupoids,being more complex,will
have higher freeenergydensity equivalents than those of the
base transitive groupoids.
A possible Morse Function for invocation of Pettini's topo
logical hypothesis or Landau's spontaneous symmetry break
ing is then a`groupoid free energy'F dened by
exp[F=H]
X
j
exp[H
G
j
=H] (8)
It is possible,using the free energyanalog F,to apply Lan
dau's spontaneous symmetry breaking arguments,and Pet
tini's topological hypothesis,to the groupoid associated with
the set of dual information sources.
Many other Morse Functions might be constructed here,
for example based on representations of the cognitive
groupoid(s).The resulting qualitative picture would not be
signicantly dierent.We will return to this argument below.
Again,Landau's and Pettini's insights regarding phase
transitions in physical systems were that certain critical phe
nomena take place in the context of a signicant alteration in
symmetry,with one phase being far more symmetric than the
other (Landau and Lifshitz,2007;Pettini,2007).A symme
try is lost in the transition { spontaneous symmetry breaking.
The greatest possible set of symmetries in a physical system
is that of the Hamiltonian describing its energy states.Usu
ally states accessible at lower temperatures will lack the sym
metries available at higher temperatures,so that the lower
temperature phase is less symmetric:The randomization of
higher temperatures ensures that higher symmetry/energy
states will then be accessible to the system.The shift be
tween symmetries is highly punctuated in the temperature
index.
The essential point is that decline in the richness of real
time environmental feedback H,or in the ability of that feed
back to in uence response,as indexed by ,can lead to punc
tuated decline in the complexity of cognitive process within
the entity of interest,according to this model.
This permits a Landauanalog phase transition analysis in
which the quality of incoming information from the embed
ding ecosystem { feedback { serves to raise or lower the pos
sible richness of an organism's cognitive response to patterns
of challenge.If H is relatively large { a rich and varied real
time environment,as perceived by the organism { then there
are many possible cognitive responses.If,however,noise or
simple constraint limit the magnitude of H,then behavior
collapses in a highly punctuated manner to a kind of ground
state in which only limited responses are possible,represented
by a simplied cognitive groupoid structure.
Certain details of such information phase transitions can be
calculated using`biological'renormalization methods (Wal
lace,2005a,Section 4.2) analogous to,but much dierent
from,those used in the determination of physical phase tran
sition universality classes (Wilson,1971).
These results represent a signicant generalization of the
Data Rate Theorem,as expressed in equation (1).
Consider,next,an inverse order parameter dened in terms
of an active attention index,a nonnegative real number R.
Thus R would be a measure of the response given to the
signal dening H.According to the Landau argument,R
disappears when H H
C
,for some critical value.That
is,when H < H
C
,there is spontaneous symmetry breaking:
only above that value can a global broadcast take place en
training numerous unconscious cognitive submodules,allow
ing R > 0.Below H
C
,no global broadcast takes place,and
attention is fragmented,or centered elsewhere,so that R = 0.
A classic Landau order parameter might be constructed as
2=(1 +exp[aR]),or 1=[1 +(aR)
n
],where a;n 1.
9 Another picture
Here we use the rich vocabulary associated with the stabil
ity of stochastic dierential equations to model,from an
other perspective,phase transitions in the composite system
of`brain/body/environment'(e.g.,Horsthemeke and Lefever,
2006;Van den Broeck et al.,1994,1997).
Dene a`symmetry entropy'based on the Morse Function
F of equation (8) over a set of structural parameters Q =
[Q
1
;:::;Q
n
] (that may include Hand other information source
uncertainties) as the Legendre transform
S = F(Q)
X
i
Q
i
@F(Q)=@Q
i
(9)
The dynamics of such a system will be driven,at least in
rst approximation,by Onsagerlike nonequilibrium thermo
dynamics relations having the standard form (de Groot and
Mazur,1984):
dQ
i
=dt =
X
j
K
i;j
@S=@Q
j
;(10)
where the K
i;j
are appropriate empirical parameters and t is
the time.A biological system involving the transmission of
information may,or may not,have local time reversibility:
in English,for example,the string`eht'has a much lower
probability than`the'.Without microreversibility,K
i;j
6=
K
j;i
.
Since,however,biological systems are quintessentially
noisy,a more tting approach is through a set of stochastic
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dierential equations having the form
dQ
i
t
= K
i
(t;Q)dt +
X
j
i;j
(t;Q)dB
j
;(11)
where the K
i
and
i;j
are appropriate functions,and dierent
kinds of`noise'dB
j
will have particular kinds of quadratic
variation aecting dynamics (Protter,1990).
Several important dynamics become evident:
1.Setting the expectation of equation (11) equal to zero
and solving for stationary points gives attractor states since
the noise terms preclude unstable states.Obtaining this re
sult,however,requires some further development.
2.This system may converge to limit cycle or pseudo
random`strange attractor'behaviors similar to thrashing in
which the system seems to chase its tail endlessly within a
limited venue { a kind of`Red Queen'pathology.
3.What is converged to in both cases is not a simple state
or limit cycle of states.Rather it is an equivalence class,or set
of them,of highly dynamic modes coupled by mutual inter
action through crosstalk and other interactions.Thus`stabil
ity'in this structure represents particular patterns of ongoing
dynamics rather than some identiable static conguration
or`answer'.These are,then,quasistationary nonequlibrium
states.
4.Applying Ito's chain rule for stochastic dierential equa
tions to the (Q
j
t
)
2
and taking expectations allows calculation
of variances.These may depend very powerfully on a system's
dening structural constants,leading to signicant instabili
ties depending on the magnitudes of the Q
i
,as in the Data
Rate Theorem (Khasminskii,2012).
5.Following the arguments of Champagnat et al.(2006),
this is very much a coevolutionary structure,where funda
mental dynamics are determined by the feedback between in
ternal and external.
In particular,setting the expectation of equation (11) to
zero generates an index theorem (Hazewinkel,2002) in the
sense of Atiah and Singer (1963),that is,an expression that
relates analytic results,the solutions of the equations,to un
derlying topological structure,the eigenmodes of a compli
cated geometric operator whose groupoid spectrumrepresents
symmetries of the possible changes that must take place for
a global workspace to become activated.
Consider,now,the attention measure,R,above.Suppose,
once triggered,the reverberation of cognitive attention to an
incoming signal is explosively selfdynamic {`reentrant'{ but
that the recognition rate is determined by the magnitude of
of the signal H,and aected by noise,so that
dR
t
= HR
t
jR
t
R
0
jdt +R
t
dW
t
(12)
where dW
t
represents white noise,and all constants are posi
tive.At nonequilibriumsteady state,the expectation of equa
tion (12) { the mean attention level { is either zero or the
canonical excitation level R
0
.
But Wilson (1971) invokes uctuation at all scales as the es
sential characteristic of physical phase transition,with invari
ance under renormalization dening universality classes.Crit
icality in biological or other cognitive systems is not likely to
be as easily classied,e.g.,Wallace (2005a,Section 4.2),but
certainly failure to have a second moment seems a good analog
to Wilson's instability criterion.As discussed above,analo
gous results relating phase transitions to noise in stochas
tic dierential equation models are widely described in the
physics literature.
To calculate the second moment in R,now invoke the Ito
chain rule,letting Y
t
= R
2
t
.Then
dY
t
= (2HjR
t
R
0
jR
2
t
+
2
R
2
t
)dt +2R
2
t
dW
t
(13)
where
2
R
2
t
in the dt term is the Ito correction due to noise.
Again taking the expectation at steady state,no second mo
ment can exist unless the expectation of R
2
t
is greater than or
equal to zero,giving the condition
H >
2
2R
0
(14)
Thus,in consonance with the direct phase transition ar
guments in H,there is a minimum signal level necessary to
support a selfdynamic attention state,in this model.The
higher the`noise'{ and the weaker the strength of the excited
state { the greater the needed environmental signal strength
to trigger punctuated`reentrant'attention dynamics.
This result,analogous to equation (1),has evident impli
cations for the quality of attention states in the context of
environmental interaction.
10 Large deviations
As Champagnat et al.(2006) describe,shifts between the
quasisteady states of a coevolutionary system like that of
equation (11) can be addressed by the large deviations formal
ism.The dynamics of drift away from trajectories predicted
by the canonical equations can be investigated by consider
ing the asymptotic of the probability of`rare events'for the
sample paths of the diusion.
`Rare events'are the diusion paths drifting far away from
the direct solutions of the canonical equation.The probability
of such rare events is governed by a large deviation principle,
driven by a`rate function'I that can be expressed in terms
of the parameters of the diusion.
This result can be used to study longtime behavior of the
diusion process when there are multiple attractive singular
ities.Under proper conditions,the most likely path followed
by the diusion when exiting a basin of attraction is the one
minimizing the rate function I over all the appropriate tra
jectories.
An essential fact of large deviations theory is that the rate
function I almost always has the canonical form
I =
X
j
P
j
log(P
j
) (15)
for some probability distribution (Dembo and Zeitouni,1998).
The argument relates to equation (11),now seen as subject
to large deviations that can themselves be described as the
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output of an information source (or sources),say L
D
,dening
I,driving Q
j
parameters that can trigger punctuated shifts
between quasisteady state topological modes of interacting
cognitive submodules.
It should be clear that both internal and feedback sig
nals,and independent,externallyimposed perturbations as
sociated with the source uncertainty I,can cause such tran
sitions in a highly punctuated manner.Some impacts may,
in such a coevolutionary system,be highly pathological over
a developmental trajectory,necessitating higher order regu
latory system counterinterventions over a subsequent trajec
tory.
Similar ideas are now common in systems biology (e.g.,Ki
tano 2004).
11 A canonical failure mode
An information source dening a large deviations rate func
tion I in equation (15) can also represent input from`unex
pected or unexplained internal dynamics'(UUID) unrelated
to external perturbation.Such UUID will always be possible
in suciently large cognitive systems,since crosstalk between
cognitive submodules is inevitable,and any possible critical
value will be exceeded if the structure is large enough or is
driven hard enough.This suggests that,as Nunney (1999) de
scribes for cancer,largescale cognitive systems must be em
bedded in powerful regulatory structures over the life course.
Wallace (2005b),in fact,examines a`cancer model'of regu
latory failure for mental disorders.
More specically,the arguments leading to equations (7)
and (8) could be reexpressed using a joint information source
H(X
G
i
;Y;L
D
) (16)
providing a more complete picture of largescale cognitive dy
namics in the presence of embedding regulatory systems,or
of sporadic external`therapeutic'interventions.However,the
joint information source of equation (16) now represents a
defacto distributed cognition involving interpenetration be
tween both the underlying embodied cognitive process and its
similarly embodied regulatory machinery.
That is,we can now dene a composite Morse Function of
embodied cognitionandregulation,F,as
exp[F=!(H;)]
X
i
exp[H(X
G
i
;Y;L
D
)=!(H;)] (17)
where!(H;) is a monotonic increasing function of both the
data rate H and of the`richness'of the internal cognitive
function dened by an internal { strictly cognitive { network
coupling parameter ,a more limited version of the argu
ment in Section 6.Typical examples might include!
0
p
H,
!
0
[H]
, > 0,!
1
log[!
2
H +1],and so on.
More generally,H(X
G
i
;Y;L
D
) in equation (17) could prob
ably be replaced by the norm
j
Y;L
D
(G
i
)j
for appropriately chosen representations of the underlying
cognitivedened groupoid,in the sense of Bos (2007) and
Buneci (2003).That is,many Morse Functions similarly pa
rameterized by the monotonic functions!(H;) are possi
ble,with the underlying topology,in the sense of Pettini,it
self more subtly parameterized,in a way,by the information
sources Y and L
D
.
Applying Pettini's topological hypothesis to the chosen
Morse Function,reduction of either H or ,or both,can
trigger a`ground state collapse'representing a phase tran
sition to a less (groupoid) symmetric`frozen'state.In
higher organisms,which must generally function under real
time constraints,elaborate secondary backup systems have
evolved to take over behavioral control under such condi
tions.These typically range across basic emotional,as well
as hypothalamicpituitaryadrenal (HPA) and hypothalamic
pituitarythyroid (HPT) axis,responses (e.g.,Wallace,2005a,
2012,2013;Wallace and Fullilove,2008).Dysfunctions of
these systems are implicated across a vast spectrum of com
mon,and usually comorbid,mental and physical disorders
(e.g.,Wallace,2005a,b;Wallace and Wallace,2010,2013).
Given the inability of some halfbillion years of evolutionary
selection pressures to successfully overcome such challenges
{ comorbid mental and physical disorders before senescence
remain rampant in human populations { it seems unlikely
that automatons or manmachine cockpits designed for the
control of critical realtime systems can avoid groundstate
collapse and other critical failure modes,if naively deployed
(e.g.,Hawley,2006,2008).Indeed,the conundrum of`robot
emotions'has already engendered considerable study (e.g.,
Fellous and Arbib,2005).
12 Discussion and conclusions
Bernard Baars'global workspace model of consciousness
(Baars,1988;Baars et al.,2013;Wallace,2005a) posits a
`theater spotlight'involving the recruitment of unconscious
cognitive modules of the brain into a temporary,tunable,gen
eral broadcast fueled by crosstalk that allows formation of the
shifting coalitions needed to address realtime problems facing
a higher organism.Similar exaptations of crosstalk between
cognitive modules at smaller scales have been recognized in
wound healing,the immune system,and so on (Wallace,2012;
2014).Newlydeveloped views of embodied cognition envision
that phenomenon as analogous,that is,as the temporary as
sembly of interacting modules from brain,body,and envi
ronment to address realtime problems facing an organism
(or a machine).This is likewise a dynamic process that sees
many available information sources { not limited to those dual
to cognitive brain or internal physiological modules { again
linked by crosstalk into a tunable realtime phenomenon that
might well be characterized as a generalized consciousness.
Here,we have made formal use of the Data Rate Theorem
in exploring the dynamics of such an embodied cognition,and
of a necessarily related embodied regulation.These,accord
ing to theory,inevitably involve a synergistic interpenetration
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among nested sets of actors,represented here as information
sources.They may include dual sources to internal cogni
tive modules,body bauplan,environmental information,lan
guage,culture,and so on.
Two factors determine the possible range of realtime cog
nitive response,in the simplest version of the model.These
are the magnitude of of the environmental feedback signal
and the inherent structural richness of the underlying cogni
tive groupoid.If that richness is lacking { if the possibility of
internal connections is limited { then even very high levels
of H may not be adequate to activate appropriate behavioral
responses to important realtime feedback signals,following
the argument of equation (17).
Cognition and regulation must,then,be viewed as inter
acting gestalt processes,involving not just an atomized indi
vidual (or,taking an even more limited`NIMH'perspective,
just the brain of that individual),but the individual in a rich
context that must include both the body that acts on the
environment,and the environment that reacts on body and
brain.
The large deviations analysis suggests that cognitive func
tion also occurs in the context,not only of a powerful en
vironmental embedding,but of a specic regulatory milieu:
there can be no cognition without regulation.The`stream of
generalized consciousness'represented by embodied cognition
must be contained within regulatory riverbanks.
For humans,and many other animal species (e.g.,Avital
and Jabolonka,2000),this picture must be expanded by an
other layer of information sources:as the evolutionary anthro
pologist Robert Boyd has expressed it,`Culture is as much a
part of human biology as the enamel on our teeth'.Thus,for
humans,the schematic hierarchy of interacting information
sources becomes
Brain!Body!Culture!Environment
Current theorizing regarding embodied cognition omits the
critical level of cultural modulation.
But matters are still more complicated.Figure 1 shows a
schematic of a`generalized consciousness'involving dynamic
patterns of crosstalk between information sources { the X
j
{
representing brain,body,culture,and environment,in no par
ticular order,and treated as fundamentally equivalent.The
full and dotted lines represent recruitment of these dispersed
resources by the organism (involving crosstalk at or above
some tunable value!) in two dierent topological patterns to
address two dierent kinds of problems in real time.
`Mental disorders',in a large sense,emerge as a synergistic
dysfunction of internal process and regulatory milieu,which
above was simply characterized by the interaction between
the driving parameters and H.Other forms of dysfunction
likely involve characteristic irregularities in topological con
nections.For example,autism spectrum and schizophreno
form disorders are widely viewed as caused by failures in
linkage that limit recruitment of unconscious cognitive brain
modules (e.g.,Wallace,2005b).Thus analogous disorders
might arise across a variety of cognitive structures from simi
Figure 1:Full and dotted lines represent two dierent re
cruitments of brain,body,cultural,and environmental infor
mation sources to address realtime problems facing an organ
ism,machine,or distributed cognition system.Both under
lying topology and the crosstalk index!(H;) are dynam
ically tunable,representing a kind of generalized conscious
ness.Pathological restrictions on connectivity or topology
would be manifest as analogs to autism or schizophrenia,in
this model,in addition to the`anxiety/depression'mode of
groundstate collapse.
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lar`topological failures'aecting the realtime recruitment of
brain or CPU system,body or eector structure,regulatory,
and environmental information sources.The central role of
culture in human biology means,of course,that,for humans,
all such disorders are inherently`culture bound syndromes',
much in the spirit of Kleinman and Cohen (1997) and Heine
(2001).
We have,in a way,extended the criticisms of Bennett and
Hacker (2003) who explored the mereological fallacy of a de
contextualization that attributes to`the brain'what is the
province of the whole individual.Here,we argue that the
`whole individual'involves essential interactions with embed
ding environmental and regulatory settings that,for humans,
must include cultural heritage and social dynamics.Realtime
automatons and manmachine cockpits,currently expected to
function without such an embedding and pervasive regulatory
milieu,may face intractable stability problems,particularly
subject to a ground state collapse in which`all possible tar
gets are enemies'.
It is clear that the BICA Challenge must take seriously the
possibility that creating a generalpurpose reallife computa
tional equivalent of the human mind using an approach based
on biologically inspired cognitive architectures will confront
the conundrum of analogs to poorlyunderstood human psy
chological disorders.We have explored one such here at length
as`ground state collapse',but,as discussed,more diverse and
subtle forms seem likely.As JohnsonLaird et al.(2006) in
dicate,surprisingly little is known about such dysfunction in
humans.For a very long time,the study of mental disorders
has been strongly dominated by a simplistic braincentered
`biological'psychiatry driven largely by the interests of the
pharmaceutical industry,which has since abandoned the ef
fort as a dry hole.The story is well known,and parallels
the arguments in Chapters 1 and 5 of Wallace and Wallace
(2013).
What is sorely needed is a cognitive theory of mental dis
orders that can apply across the many possible underlying
modalities,be they biological,biopsychosociocultural,in sil
ico,or entities of distributed cognition,ranging from man
machine cockpits to largescale social enterprises.At present,
virtually no such work is actively supported.
Within the next few decades,even discounting BICA eort,
machine entities having truly massive numbers of computing
cores will be tasked with the realtime control of many com
plex critical processes under fogofwar conditions.We would
do well to devote some preliminary thought to what an en
gineering discipline of`machine psychiatry'for such systems
might look like.
13 Mathematical appendix
13.1 Morse Theory
Morse Theory explores relations between analytic behavior of
a function { the location and character of its critical points
{ and the underlying topology of the manifold on which the
function is dened.We are interested in a number of such
functions,for example information source uncertainty on a
parameter space and possible iterations involving parameter
manifolds determining critical behavior.An example might
be the sudden onset of a giant component.These can be re
formulated from a Morse Theory perspective (Pettini,2007).
The basic idea of Morse Theory is to examine an n
dimensional manifold M as decomposed into level sets of some
function f:M!R where R is the set of real numbers.The
alevel set of f is dened as
f
1
(a) = fx 2 M:f(x) = ag;
the set of all points in M with f(x) = a.If M is compact,then
the whole manifold can be decomposed into such slices in a
canonical fashion between two limits,dened by the minimum
and maximum of f on M.Let the part of M below a be
dened as
M
a
= f
1
(1;a] = fx 2 M:f(x) ag:
These sets describe the whole manifold as a varies between
the minimum and maximum of f.
Morse functions are dened as a particular set of smooth
functions f:M!R as follows.Suppose a function f has
a critical point x
c
,so that the derivative df(x
c
) = 0,with
critical value f(x
c
).Then,f is a Morse function if its critical
points are nondegenerate in the sense that the Hessian matrix
of second derivatives at x
c
,whose elements,in terms of local
coordinates are
H
i;j
= @
2
f=@x
i
@x
j
;
has rank n,which means that it has only nonzero eigenvalues,
so that there are no lines or surfaces of critical points and,
ultimately,critical points are isolated.
The index of the critical point is the number of negative
eigenvalues of H at x
c
.
A level set f
1
(a) of f is called a critical level if a is a
critical value of f,that is,if there is at least one critical point
x
c
2 f
1
(a).
Again following Pettini (2007),the essential results of
Morse Theory are:
1.If an interval [a;b] contains no critical values of f,then
the topology of f
1
[a;v] does not change for any v 2 (a;b].
Importantly,the result is valid even if f is not a Morse func
tion,but only a smooth function.
2.If the interval [a;b] contains critical values,the topology
of f
1
[a;v] changes in a manner determined by the properties
of the matrix H at the critical points.
3.If f:M!R is a Morse function,the set of all the
critical points of f is a discrete subset of M,i.e.,critical
points are isolated.This is Sard's Theorem.
4.If f:M!Ris a Morse function,with M compact,then
on a nite interval [a;b] R,there is only a nite number of
critical points p of f such that f(p) 2 [a;b].The set of critical
values of f is a discrete set of R.
5.For any dierentiable manifold M,the set of Morse
functions on M is an open dense set in the set of real functions
of M of dierentiability class r for 0 r 1.
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6.Some topological invariants of M,that is,quantities that
are the same for all the manifolds that have the same topology
as M,can be estimated and sometimes computed exactly once
all the critical points of f are known:let the Morse numbers
i
(i = 0;:::;m) of a function f on M be the number of critical
points of f of index i,(the number of negative eigenvalues of
H).The Euler characteristic of the complicated manifold M
can be expressed as the alternating sumof the Morse numbers
of any Morse function on M,
=
m
X
i=1
(1)
i
i
:
The Euler characteristic reduces,in the case of a simple poly
hedron,to
= V E +F
where V;E,and F are the numbers of vertices,edges,and
faces in the polyhedron.
7.Another important theorem states that,if the interval
[a;b] contains a critical value of f with a single critical point
x
c
,then the topology of the set M
b
dened above diers from
that of M
a
in a way which is determined by the index,i,of
the critical point.Then M
b
is homeomorphic to the manifold
obtained from attaching to M
a
an ihandle,i.e.,the direct
product of an idisk and an (mi)disk.
Pettini (2007) and Matsumoto (2002) contain details and
further references.
13.2 Groupoids
A groupoid,G,is dened by a base set A upon which some
mapping { a morphism { can be dened.Note that not
all possible pairs of states (a
j
;a
k
) in the base set A can be
connected by such a morphism.Those that can dene the
groupoid element,a morphism g = (a
j
;a
k
) having the natu
ral inverse g
1
= (a
k
;a
j
).Given such a pairing,it is possi
ble to dene`natural'endpoint maps (g) = a
j
;(g) = a
k
from the set of morphisms G into A,and a formally as
sociative product in the groupoid g
1
g
2
provided (g
1
g
2
) =
(g
1
);(g
1
g
2
) = (g
2
),and (g
1
) = (g
2
).Then,the prod
uct is dened,and associative,(g
1
g
2
)g
3
= g
1
(g
2
g
3
).In addi
tion,there are natural left and right identity elements
g
;
g
such that
g
g = g = g
g
.
An orbit of the groupoid G over A is an equivalence class
for the relation a
j
Ga
k
if and only if there is a groupoid
element g with (g) = a
j
and (g) = a
k
.A groupoid is called
transitive if it has just one orbit.The transitive groupoids
are the building blocks of groupoids in that there is a natural
decomposition of the base space of a general groupoid into
orbits.Over each orbit there is a transitive groupoid,and
the disjoint union of these transitive groupoids is the original
groupoid.Conversely,the disjoint union of groupoids is itself
a groupoid.
The isotropy group of a 2 X consists of those g in G with
(g) = a = (g).These groups prove fundamental to classi
fying groupoids.
If G is any groupoid over A,the map (;):G!AA is
a morphism from G to the pair groupoid of A.The image of
(;) is the orbit equivalence relation G,and the functional
kernel is the union of the isotropy groups.If f:X!Y is a
function,then the kernel of f,ker(f) = [(x
1
;x
2
) 2 X X:
f(x
1
) = f(x
2
)] denes an equivalence relation.
Groupoids may have additional structure.For example,a
groupoid G is a topological groupoid over a base space X if
G and X are topological spaces and ; and multiplication
are continuous maps.
In essence,a groupoid is a category in which all morphisms
have an inverse,here dened in terms of connection to a base
point by a meaningful path of an information source dual to
a cognitive process.
The morphism (;) suggests another way of looking at
groupoids.A groupoid over A identies not only which ele
ments of A are equivalent to one another (isomorphic),but it
also parameterizes the dierent ways (isomorphisms) in which
two elements can be equivalent,i.e.,in our context,all possible
information sources dual to some cognitive process.Given the
information theoretic characterization of cognition presented
above,this produces a full modular cognitive network in a
highly natural manner.
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